Problem: Simplify; express your answer in exponential form. Assume $z\neq 0, p\neq 0$. $\dfrac{{(z^{-1}p)^{-5}}}{{(z^{5}p^{3})^{2}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(z^{-1}p)^{-5} = (z^{-1})^{-5}(p)^{-5}}$ On the left, we have ${z^{-1}}$ to the exponent ${-5}$ . Now ${-1 \times -5 = 5}$ , so ${(z^{-1})^{-5} = z^{5}}$ Apply the ideas above to simplify the equation. $\dfrac{{(z^{-1}p)^{-5}}}{{(z^{5}p^{3})^{2}}} = \dfrac{{z^{5}p^{-5}}}{{z^{10}p^{6}}}$ Break up the equation by variable and simplify. $\dfrac{{z^{5}p^{-5}}}{{z^{10}p^{6}}} = \dfrac{{z^{5}}}{{z^{10}}} \cdot \dfrac{{p^{-5}}}{{p^{6}}} = z^{{5} - {10}} \cdot p^{{-5} - {6}} = z^{-5}p^{-11}$